308 XV NORMED ALGEBRAS AND SPECTRAL THEORY
Remark
(15.1.8) An algebra A over C, endowed with a topology, is said to b
normable if its topology can be defined by a norm with respect to which £
becomes a normed algebra in the sense defined above. For this it is necessar
and sufficient that the topology of A should be definable by a norm, anc
should be compatible with the vector space structure of A, and that the map-
ping (x, y)\-+xy of A x A into A should be continuous for this topology
Clearly these conditions are necessary. To see they are sufficient, observe thai
if \\x\\ is a norm defining the topology of A, then there exists a constam
a > 0 such that \\xy\\ g, a - \\x\\ • \\y\\ for all x, y in A (5.5.1). If we replace
the norm \\x\\ by the equivalent norm (5.6.1) a||jc|| = \\x\\i9 then clearl}
||jcy|li ^ Wi. • lljlli- This establishes our assertion in the case where A has nc
unit element. If A has a unit element e ^ 0, we consider for each x e A the
left translation Lx:y^xy; this is a continuous endomorphism of the
normed space A, and we have Lx+x> = Lx + Lx> and L^x = 1LX for all 1 e C.
and Lxx> =LX° Lx>. In other words, the mapping x\~>Lx is a homomorphism
of the algebra A into the algebra j£?(A) of continuous endomorphisms of the
normed space A. Moreover, this homomorphism is infective, because the
relation Lx = 0 implies x = xe = Lx(e) = 0. Now we have \\LX\\ <; a \\x\\ (5.7)
and on the other hand, ||jc|| = \\xe\\ g \\LX\\ • \\e\\. It follows that if we put
11-^112 = IIAcll? then \\x\\2 is a norm on A which is equivalent to the given
norm ||,x|| (5.6.1), and for which A is a normed algebra.
PROBLEMS
1. Let A be the C-algebra of complex-valued functions defined and k times continuously
differentiable on [0, 1]. Show that the function
1*11 = t A SUP |X<»<0|
A = 0 hi O^r^i
on A is a norm for which A is a Banach algebra.
2. Let (an) be a sequence of real numbers >0 such that a0= 1, am+n^awan and
00
lim ai/n = 0. Let A denote the set of all formal power series x = ^ £nTn with the
n-»oo n = 0
oo
complex coefficients frt such that J) a«|f«| < + oo. Show that A is a subalgebra of the
n = 0
00
algebra C[[T]] of formal power series in T, and that \\x\\ = X anl£»l is a norm on A
nsO
for which A is a Banach algebra with unit element. is not, in general, an ideal (unless G is compact